How to make a texture interact with scene lighting and shadows in cycles
You do not need the Transparent BSDF node. Just plug the alpha from the image texture into the Fac of the Mix node.

The Transparent BSDF is a shader node, and the Fac on the Mix node is looking for a value 0 through 1, a black and white map.

Your logo will "interact with light", it just will not be glossy because of the way you are mixing the glossy nodes.

This is a simple node setup to mix the diffuse logo with a glossy node, yet still have the logo partially glossy.
Use the "white" end of the ColorRamp node to control how glossy the logo will be. White (a value of 1) will have no gloss, and black (a value of 0) will be full gloss, and you would not see the logo.